math review

Math ReviewMath Review

Algebra and Functions ReviewAlgebra and Functions Review

Operations on Algebraic Operations on Algebraic ExpressionsExpressions

You will need to be able to apply the basic You will need to be able to apply the basic operations of arithmetic – addition, operations of arithmetic – addition, subtraction, multiplication, and division - subtraction, multiplication, and division - to algebraic expressionsto algebraic expressions In examples like this:In examples like this:

4x+5x=9x4x+5x=9x

10z-3y-(-2z)+2y = 12z-y10z-3y-(-2z)+2y = 12z-y

(x+3)(x-2) = x^2+x-6(x+3)(x-2) = x^2+x-6

FactoringFactoring

Difference of 2 squaresDifference of 2 squares a^2-b^2 = (a+b)(a-b)a^2-b^2 = (a+b)(a-b)

Finding common factorsFinding common factors x^2+2x = x(x+2)x^2+2x = x(x+2) 2x+4y = 2(x+2y)2x+4y = 2(x+2y)

Factoring quadraticsFactoring quadratics x^2-3x-4 = (x-4)(x+1)x^2-3x-4 = (x-4)(x+1) X^2+2x+1 = (x+1)(x+1) = (x+1)^2X^2+2x+1 = (x+1)(x+1) = (x+1)^2

ExponentsExponents

DefinitionsDefinitions a^3 = a x a x aa^3 = a x a x a P^-4 = 1/P x 1/P x 1/P x 1/PP^-4 = 1/P x 1/P x 1/P x 1/P X^0 = 1X^0 = 1 X ^ (a/b) = (b square root of x)^aX ^ (a/b) = (b square root of x)^a Y^ (1/2) = (square root of y)Y^ (1/2) = (square root of y)

ExponentsExponents

Three Points to RememberThree Points to Remember When multiplying expressions with the same base, When multiplying expressions with the same base,

add the exponentsadd the exponentsa^2 * a^5 = a^7a^2 * a^5 = a^7t^5 * t^-2 = t^3t^5 * t^-2 = t^3

When dividing expressions with the same base, When dividing expressions with the same base, subtract exponentssubtract exponents

r^5/r^3 = r^2r^5/r^3 = r^2a^m/a^n = a^ (m-n)a^m/a^n = a^ (m-n)

When a number raised to an exponent is raised to a When a number raised to an exponent is raised to a second exponent, multiply the exponentssecond exponent, multiply the exponents

(n^3)^6 = n^18(n^3)^6 = n^18(a^m)^n = a^mn (a^m)^n = a^mn

Evaluating Expressions with Evaluating Expressions with Exponents and RootsExponents and Roots

You will need to know how to evaluate You will need to know how to evaluate expressions involving exponents and rootsexpressions involving exponents and roots If y = 8, what is y^(2/3)If y = 8, what is y^(2/3)

y^(2/3) = 3y^(2/3) = 3rdrd square root of 8^2 = 4 square root of 8^2 = 4 If x^(3/2) = 64, what is the value of x?If x^(3/2) = 64, what is the value of x?

x^(3/2) = (x^3)^1/2 = 64x^(3/2) = (x^3)^1/2 = 64

Cube both sides, and x^1/2 = 4 Cube both sides, and x^1/2 = 4

Square both sides and x = 16 Square both sides and x = 16

Solving EquationsSolving Equations

Working with “Unsolvable” EquationsWorking with “Unsolvable” Equations Though you can’t solve the equation, you can Though you can’t solve the equation, you can

answer the question.answer the question.If a+b=5, what is the value of 2a+2b?If a+b=5, what is the value of 2a+2b?

You can’t solve for a or b, but it doesn’t ask thatYou can’t solve for a or b, but it doesn’t ask that

Factor: 2(a+b), so 2(5) = 10Factor: 2(a+b), so 2(5) = 10


Solving for One Variable in Terms of Solving for One Variable in Terms of AnotherAnother You will not always be able to find a specific, You will not always be able to find a specific,

numerical value for all the variables, but you numerical value for all the variables, but you can still solve for one variable in terms of can still solve for one variable in terms of another oneanother one

If 3x+y= z, what is x in terms of y and z?If 3x+y= z, what is x in terms of y and z? Put X on one side of equation by itselfPut X on one side of equation by itself 3x=z-y, then x = (z-y)/33x=z-y, then x = (z-y)/3


Involving Radical ExpressionsInvolving Radical Expressions 5(square root of x) is a radical expression 5(square root of x) is a radical expression

because it involves a rootbecause it involves a root A radical equation is one that involves a A radical equation is one that involves a

radical expressionradical expression5(square root of x) +14 = 295(square root of x) +14 = 29

5(square root of x) = 155(square root of x) = 15

(square root of x)=3(square root of x)=3

X=9X=9

Absolute ValueAbsolute Value

Absolute Value of a number is its distance Absolute Value of a number is its distance from zero on the number line. Denoted: IxIfrom zero on the number line. Denoted: IxI I6.5I=6.5, I-32I=32I6.5I=6.5, I-32I=32 Think of it as its “size” of the number, Think of it as its “size” of the number,

disregarding pos or negdisregarding pos or neg ExampleExample

I7-tI=10I7-tI=10 Can be split up into two equations:Can be split up into two equations:

7-t=10 and –(7-t)=107-t=10 and –(7-t)=10t=-3 or t=17 t=-3 or t=17

Direct Translation into Direct Translation into Mathematical ExpressionsMathematical Expressions

Many word problems require you to translate the Many word problems require you to translate the verbal description of a mathematical fact or verbal description of a mathematical fact or relationship into mathematical terms. relationship into mathematical terms.

Always read the word problem carefully and Always read the word problem carefully and double-check that you have translated it exactly.double-check that you have translated it exactly. ““3 times the quantity” (4x+6) is 3(4x+6)3 times the quantity” (4x+6) is 3(4x+6) ““A number y decreased by 60” is y-60A number y decreased by 60” is y-60 ““20 divided by n” is (20/n)20 divided by n” is (20/n) ““20 divided into a number y” is (y/20)20 divided into a number y” is (y/20)

InequalitiesInequalities

An inequality is a statement that one quantity is An inequality is a statement that one quantity is greater than or less than anothergreater than or less than another

Symbols:Symbols: Greater than: > (5>3)Greater than: > (5>3) Less than: < (-7 < -6)Less than: < (-7 < -6) A line under the symbol means “or equal to”A line under the symbol means “or equal to”

Example:Example: 2x+1>112x+1>11

2x>11-1, 2x>102x>11-1, 2x>10

So, x>5So, x>5

Systems of Linear Equations and Systems of Linear Equations and InequalitiesInequalities

You may be asked to solve systems of two You may be asked to solve systems of two or more linear equations or inequalities. or more linear equations or inequalities.

Example: Example: For what values of a and b are the following For what values of a and b are the following

equations true? a+2b=1, -3a-8b=1equations true? a+2b=1, -3a-8b=1Eliminate one of the variables, let’s do b. so Eliminate one of the variables, let’s do b. so multiply both sides by 4 to get the b’s equal.multiply both sides by 4 to get the b’s equal.

4a+8b=4 + -3a-8b=14a+8b=4 + -3a-8b=1 Add the two equations: a=5, then put back into first Add the two equations: a=5, then put back into first

equation and you get b=-2equation and you get b=-2

Solving Quadratic Equations by Solving Quadratic Equations by FactoringFactoring

You may be asked to solve quadratic You may be asked to solve quadratic equations that can be factored. You will equations that can be factored. You will not be asked to use the quadratic formula.not be asked to use the quadratic formula.

Example:Example: For what values of x is x^2-10x+20=-4For what values of x is x^2-10x+20=-4

Add 4 to both sides to get the standard quad. Add 4 to both sides to get the standard quad. Formula: x^2-10x+24=0Formula: x^2-10x+24=0

Now factor: (x-4)(x-6)=0Now factor: (x-4)(x-6)=0

So, either x-4=0 and x-6=0, so x=4 and x=6So, either x-4=0 and x-6=0, so x=4 and x=6

Rational Equations and InequalitiesRational Equations and Inequalities

A rational algebraic expression is the A rational algebraic expression is the quotient of two polynomials. quotient of two polynomials.

Example:Example: For what value of x is the following equation For what value of x is the following equation

true? 3=(x-1)/(2x+3)true? 3=(x-1)/(2x+3) Multiply both sides by (2x+3)Multiply both sides by (2x+3)

6x+9=x-16x+9=x-1 5x=-105x=-10 X=-2X=-2

Direct and Inverse VariationDirect and Inverse Variation

The quantities x and y are directly proportional if The quantities x and y are directly proportional if y=kx for some constant k.y=kx for some constant k. If x and y are directly proportional, when x is 10, y is If x and y are directly proportional, when x is 10, y is

equal to -5. If x=3, then what is y?equal to -5. If x=3, then what is y?y=kx, so solve for k with 10 and -5y=kx, so solve for k with 10 and -5

-5=k(10), k=(-1/2)-5=k(10), k=(-1/2) So then plug in k with 3: y=3(-1/2)So then plug in k with 3: y=3(-1/2)

y=(-3/2)y=(-3/2)

Inversely proportional: y=k/xInversely proportional: y=k/x If xy=4, show the x and y are inversely proportionalIf xy=4, show the x and y are inversely proportional

So y=4/x, so with k as 4, then they are inversely prop.So y=4/x, so with k as 4, then they are inversely prop.

Word ProblemsWord Problems

Some math questions are presented as word problems Some math questions are presented as word problems to apply math skills to everyday situations. With word to apply math skills to everyday situations. With word problems you need to:problems you need to:

Read and interpret what is being askedRead and interpret what is being asked Determine what information you are givenDetermine what information you are given Determine what information you need to knowDetermine what information you need to know Decide what mathematical skills or formulas you need to apply Decide what mathematical skills or formulas you need to apply

to find the answerto find the answer Work out the answerWork out the answer Double check to make sure the answer makes sense. When Double check to make sure the answer makes sense. When

checking word problems, don’t substitute your answer into your checking word problems, don’t substitute your answer into your equations, because they may be wrong. Instead, check word equations, because they may be wrong. Instead, check word problems by checking your answer with the original words.problems by checking your answer with the original words.


As you read the problems, translate the As you read the problems, translate the words into mathematical expressions and words into mathematical expressions and equations. equations. Is, Was, Has means = (equals)Is, Was, Has means = (equals) More than, older than, farther than, greater More than, older than, farther than, greater

than, sum of means + (addition)than, sum of means + (addition) Less than, difference, younger than, fewer Less than, difference, younger than, fewer

means – (subtraction)means – (subtraction) Of means x (multiplication, percent)Of means x (multiplication, percent) For, per means / (ratio or division)For, per means / (ratio or division)


ExampleExample The price of a sweater went up 20% since last The price of a sweater went up 20% since last

year. If last year’s price was x, what is this year. If last year’s price was x, what is this year’s price in terms of x?year’s price in terms of x?

So, last year’s price is 100% of xSo, last year’s price is 100% of x

This year’s price is 100% of x plus 20% of xThis year’s price is 100% of x plus 20% of x x+(20% * x) = x+.2x = 1.2xx+(20% * x) = x+.2x = 1.2x

FunctionsFunctions

Function Notation and EvaluationFunction Notation and Evaluation A function can be thought of as a rule or formula that A function can be thought of as a rule or formula that

tells how to associate the elements in one set (or the tells how to associate the elements in one set (or the domain) with the elements in another set (or the domain) with the elements in another set (or the range)range)

Example: The “Squaring Function” can be thought of Example: The “Squaring Function” can be thought of as the rule “taking the square of x” or the rule x^2. as the rule “taking the square of x” or the rule x^2.

Function notation lets you write complicated functions Function notation lets you write complicated functions much more easily. much more easily.

g is defined by g(x)=3^x+(1/x). Then g(2)=3^2+(1/2)= 9 (1/2)g is defined by g(x)=3^x+(1/x). Then g(2)=3^2+(1/2)= 9 (1/2)

FunctionsFunctions

Domain and RangeDomain and Range The domain of a function is the set of all the values for which the The domain of a function is the set of all the values for which the

functions is defined.functions is defined. The range of a function is the set of all values that are the The range of a function is the set of all values that are the

output, or result, of applying the function.output, or result, of applying the function. Example: What are the domain and range of f(x)=1+(square root Example: What are the domain and range of f(x)=1+(square root

of x)of x)Domain of f is the set of all values of x for which the formula is Domain of f is the set of all values of x for which the formula is defined. This formula makes sense if x is 0 or a positive number defined. This formula makes sense if x is 0 or a positive number (can’t be negative): so domain is all nonnegative numbers x(can’t be negative): so domain is all nonnegative numbers xRange would be the set of all possible values of the equation. It Range would be the set of all possible values of the equation. It must be at least 1 (b/c of the square root). Is r greater than or equal must be at least 1 (b/c of the square root). Is r greater than or equal to 1, though? See if it could be higher:to 1, though? See if it could be higher:

1+(square root of x)=9, (square root of x)=8, x=64, so it would be 1+(square root of x)=9, (square root of x)=8, x=64, so it would be greater than or equal to 1greater than or equal to 1

Using New DefinitionsUsing New Definitions

For functions, especially those involving more than one For functions, especially those involving more than one variable, a special symbol is sometimes introduced and variable, a special symbol is sometimes introduced and defined. They will have unusual looking signs to not be defined. They will have unusual looking signs to not be confused with standard mathematical symbols. confused with standard mathematical symbols.

The key to these questions is to make sure you read the The key to these questions is to make sure you read the definition carefully.definition carefully.

Example:Example: = wy-xz, where w,y,x, and z are integers.= wy-xz, where w,y,x, and z are integers.

What is the value of What is the value of

ww xx

zz yy22 33

44 11

Functions as ModelsFunctions as Models

Functions can be used as models of real Functions can be used as models of real life situations. life situations. The temp. in City X is W(t) degrees The temp. in City X is W(t) degrees

Fahrenheit t hours after sundown at 5 pm. Fahrenheit t hours after sundown at 5 pm. The function W(t) is given by:The function W(t) is given by:

W(t)=.1(400-40t+t^2) for 0 is less than or equal to t W(t)=.1(400-40t+t^2) for 0 is less than or equal to t is less than or equal to 12.is less than or equal to 12.

Linear Functions: Their Equations Linear Functions: Their Equations and Graphsand Graphs

Linear Function: y=mx+b, where m and b Linear Function: y=mx+b, where m and b are constantsare constants m is the slope, and y intercept is bm is the slope, and y intercept is b

Quadratic Functions: Their Quadratic Functions: Their Equations and GraphsEquations and Graphs

y=x^2-1y=x^2-1

Graphs will look like a u (going up or Graphs will look like a u (going up or down). A negative answer (once solved) down). A negative answer (once solved) will go down and positive will go up. will go down and positive will go up. Largest number of y will occur when x Largest number of y will occur when x equals zero. equals zero.

Qualitative Behavior of Graphs and Qualitative Behavior of Graphs and FunctionsFunctions

Know how properties of a function and its Know how properties of a function and its graph are related.graph are related.

The zeros of the function f are given by the The zeros of the function f are given by the points where the graph of f(x) in the xy points where the graph of f(x) in the xy plane intersects the x axis. plane intersects the x axis.

Translations and their Effects on Translations and their Effects on Graphs of FunctionsGraphs of Functions

If you are given the graph g(x) you should If you are given the graph g(x) you should be able to identify the graph of g(x+3)be able to identify the graph of g(x+3) You can plug in numbers to find out specific You can plug in numbers to find out specific

points.points.

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